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Pointwise pinched manifolds are space forms. (English) Zbl 0587.53042

Geometric measure theory and the calculus of variations, Proc. Summer Inst., Arcata/Calif. 1984, Proc. Symp. Pure Math. 44, 307-328 (1986).
[For the entire collection see Zbl 0577.00014.]
Theorem 1. If the unnormalized Ricci flow \(\partial g/\partial t=-2\cdot ric\) starts at a metric with positive curvature operator, then the curvature operator stays positive. Corollary to theorem 2. If \(M\) has positive sectional curvatures which are pointwise \(\delta\) (n) pinched, then the normalized Ricci flow converges to a metric of constant sectional curvature \((\delta (n)=1-n^{-3/2}\) suffices for large \(n\)).
The estimates are based on splitting tensors with curvature tensor symmetries into their irreducible components and then on following Hamilton’s procedure with improved estimates.
Reviewer: H.Karcher

MSC:

53C20 Global Riemannian geometry, including pinching
58J35 Heat and other parabolic equation methods for PDEs on manifolds

Citations:

Zbl 0577.00014